The smallest one is given, so you start by showing that 10001 is divisible by 137.
Suppose now that for some four digit number (possibly padded by zeros from the left) A, the assumption is true, that is the number obtained by writing A twice in a row is divisible by 137. Can you write a formula for this number? Then the induction step consists of adding 1 to A and showing that the difference is divisible by 137.